Now if the terms "continuous", "in contact", and "in succession" are
understood as defined above things being "continuous" if their
extremities are one, "in contact" if their extremities are together,
and "in succession" if there is nothing of their own kind intermediate
between them-nothing that is continuous can be composed "of
indivisibles": e.g. a line cannot be composed of points, the line
being continuous and the point indivisible. For the extremities of two
points can neither be one (since of an indivisible there can be no
extremity as distinct from some other part) nor together (since that
which has no parts can have no extremity, the extremity and the
thing of which it is the extremity being distinct).

    Moreover, if that which is continuous is composed of points, these
points must be either continuous or in contact with one another: and
the same reasoning applies in the case of all indivisibles. Now for
the reason given above they cannot be continuous: and one thing can be
in contact with another only if whole is in contact with whole or part
with part or part with whole. But since indivisibles have no parts,
they must be in contact with one another as whole with whole. And if
they are in contact with one another as whole with whole, they will
not be continuous: for that which is continuous has distinct parts:
and these parts into which it is divisible are different in this
way, i.e. spatially separate.

    Nor, again, can a point be in succession to a point or a moment to a
moment in such a way that length can be composed of points or time
of moments: for things are in succession if there is nothing of
their own kind intermediate between them, whereas that which is
intermediate between points is always a line and that which is
intermediate between moments is always a period of time.

    Again, if length and time could thus be composed of indivisibles,
they could be divided into indivisibles, since each is divisible
into the parts of which it is composed. But, as we saw, no
continuous thing is divisible into things without parts. Nor can there
be anything of any other kind intermediate between the parts or
between the moments: for if there could be any such thing it is
clear that it must be either indivisible or divisible, and if it is
divisible, it must be divisible either into indivisibles or into
divisibles that are infinitely divisible, in which case it is
continuous.

    Moreover, it is plain that everything continuous is divisible into
divisibles that are infinitely divisible: for if it were divisible
into indivisibles, we should have an indivisible in contact with an
indivisible, since the extremities of things that are continuous
with one another are one and are in contact.

    The same reasoning applies equally to magnitude, to time, and to
motion: either all of these are composed of indivisibles and are
divisible into indivisibles, or none. This may be made clear as
follows. If a magnitude is composed of indivisibles, the motion over
that magnitude must be composed of corresponding indivisible
motions: e.g. if the magnitude ABG is composed of the indivisibles
A, B, G, each corresponding part of the motion DEZ of O over ABG is
indivisible. Therefore, since where there is motion there must be
something that is in motion, and where there is something in motion
there must be motion, therefore the being-moved will also be
composed of indivisibles. So O traversed A when its motion was D, B
when its motion was E, and G similarly when its motion was Z. Now a
thing that is in motion from one place to another cannot at the moment
when it was in motion both be in motion and at the same time have
completed its motion at the place to which it was in motion: e.g. if a
man is walking to Thebes, he cannot be walking to Thebes and at the
same time have completed his walk to Thebes: and, as we saw, O
traverses a the partless section A in virtue of the presence of the
motion D. Consequently, if O actually passed through A after being
in process of passing through, the motion must be divisible: for at
the time when O was passing through, it neither was at rest nor had
completed its passage but was in an intermediate state: while if it is
passing through and has completed its passage at the same moment, then
that which is walking will at the moment when it is walking have
completed its walk and will be in the place to which it is walking;
that is to say, it will have completed its motion at the place to
which it is in motion. And if a thing is in motion over the whole
KBG and its motion is the three D, E, and Z, and if it is not in
motion at all over the partless section A but has completed its motion
over it, then the motion will consist not of motions but of starts,
and will take place by a thing's having completed a motion without
being in motion: for on this assumption it has completed its passage
through A without passing through it. So it will be possible for a
thing to have completed a walk without ever walking: for on this
assumption it has completed a walk over a particular distance
without walking over that distance. Since, then, everything must be
either at rest or in motion, and O is therefore at rest in each of the
sections A, B, and G, it follows that a thing can be continuously at
rest and at the same time in motion: for, as we saw, O is in motion
over the whole ABG and at rest in any part (and consequently in the
whole) of it. Moreover, if the indivisibles composing DEZ are motions,
it would be possible for a thing in spite of the presence in it of
motion to be not in motion but at rest, while if they are not motions,
it would be possible for motion to be composed of something other than
motions.

    And if length and motion are thus indivisible, it is neither more
nor less necessary that time also be similarly indivisible, that is to
say be composed of indivisible moments: for if the whole distance is
divisible and an equal velocity will cause a thing to pass through
less of it in less time, the time must also be divisible, and
conversely, if the time in which a thing is carried over the section A
is divisible, this section A must also be divisible.